Lecture 4 Section 1.4.3 Wed, Sep 6, 2006 What’s in the Bag? Lecture 4 Section 1.4.3 Wed, Sep 6, 2006

How Strong is the Evidence? Rather than give an accept/reject answer, we may ask a different question: How strong is the evidence against H0? We use the p-value to measure this.

The p-value In the two bags, if the selected token is worth $50, what is the p-value?

Two Bags -1000 10 20 30 40 60 1000 50 Bag A Bag B

Two Bags Bag A Bag B At least as extreme as 50 -1000 10 20 30 40 50 60

Two Bags p-value = 2/20 = 0.10 Bag A Bag B At least as extreme as 50 -1000 10 20 30 40 50 60 1000 Bag B -1000 10 20 30 40 50 60 1000

The p-value If the selected token is worth $30, what is the p-value? Keep in mind, we may always compute the p-value regardless of our decision about which hypothesis to accept.

A Two-Sided Test 1 2 3 4 6 5 Bag E 8 10 9 7 Bag F 1 2 3 4 5 6 7 8 9 10

A Two-Sided Test If the selected token is worth $8, what is the p-value? First, what is the direction of extreme? Which values are at least as extreme as 8? 1 2 3 4 6 5 Bag E 8 10 9 7

A Two-Sided Test 1 2 3 4 6 5 Bag E 8 10 9 7 Bag F 1 2 3 4 5 6 7 8 9 10

A Two-Sided Test Bag E Bag F At least as extreme as 8 1 2 3 4 5 6 7 8 9 10 Bag F 1 2 3 4 5 6 7 8 9 10

A Two-Sided Test p-value = 12/30 = 0.40 Bag E Bag F 1 2 3 4 5 6 7 8 9 10 Bag F 1 2 3 4 5 6 7 8 9 10

The p-value If the selected token is worth $1, what is the p-value?

A Two-Sided Test In a two-sided test, if the null distribution is symmetric, then you can compute the probability in one direction, and then double it to get the p-value.

The p-value A small p-value is strong evidence against the null hypothesis. Why? A large p-value is evidence in favor of the null hypothesis.

The p-value In all other areas of life, large things are more significant and small things are less significant. But in statistics A small p-value is statistically significant. A large p-value is not statistically significant.

Two Explanations of Unusual Observations The null hypothesis leads us to a certain expectation of what the data will show. If the data deviate from our expectation, then we need to explain that deviation. Two explanations: The null hypothesis is true; the deviation is due to chance. The null hypothesis is false; we had the wrong expectation in the first place.

Two Explanations of Unusual Observations The first is the better explanation if the deviation is small. The second is the better explanation if the deviation is large. That’s where the p-value comes in.

Two Explanations of Unusual Observations The p-value measures the likelihood of a deviation as large as the one we observed if the null hypothesis is true. Small deviations are likely. Large deviations are unlikely. Therefore, Small deviations have large p-values. Large deviations have small p-values.

The p-value Prevalence and Cardiovascular Disease Correlates of Low Cardiorespiratory Fitness in Adolescents and Adults

Example Consider the Intelligent Design hypothesis vs. the Evolution hypothesis. Which hypothesis uses the “chance” explanation? Which hypothesis uses a non-random, directed mechanism as the explanation? Which would be the null hypothesis? Which would have the burden of proof?